Apparatus for the measurement of gravitational fields

ABSTRACT

Apparatus for measuring gravitational fields comprising a superconducting string (1) fixed at both ends and forming part of a closed superconducting loop inductively coupled to two driving solenoids (L d1 , L d2 ). Displacement of the string in response to a gravitational field is sensed by two magnetic flux transformers each comprising a signal coil and two pick-up coils ((L p1 , L p2 ). Pairs of pick-up coils lie in two perpendicular planes providing two independent channels of measurements. The two arms of each flux transformer are balanced to convert only the amplitudes of the string&#39;s antisymmetric natural modes into an output voltage. The output voltage of each channel is used to produce a feed-back current distribution (L y1 , L y2 ) proximate and parallel to the string. By adjusting the feed-back current, the effective relaxation time and resonant frequency of the first antisymmetric mode of the string can be adjusted, while leaving the symmetric modes unchanged, thus increasing the apparatus&#39; sensitivity to gravity gradients.

CROSS-REFERENCE TO RELATED APPLICATIONS

This is the national stage of International Application No.PCT/GB95/02349 filed Oct. 4, 1995.

This invention relates to the measurement of gravitational fields,particularly to gravity gradiometry, and more particularly to a methodfor measuring absolutely off-diagonal components of the gravity gradienttensor.

The gravity gradient tensor is a two-dimensional matrix of the secondpartial derivatives of a gravitational potential, V, with respect to theCartesian co-ordinates, x, y, z, of some arbitrary reference frame. Itrepresents how the gravity vector itself in each of these directionsvaries along the axes.

Accurate absolute measurements of the components of the gravity gradienttensor Γ_(ij) =∂² _(ij) V (ij=x,y,z), taken at some local coordinateframe OXYZ are very important to progress in the fields of geologicalprospecting, mapping of the Earth's gravitational field, and space, seaand underwater navigation.

A method of absolute measurement of gravity gradient tensor componentswas invented first by Baron Roland von Eotvos as early as 1890,utilising a torsion balance with proof masses hung at different heightsfrom a horizontal beam suspended by a fine filament. The gravitygradients give rise to differential forces being applied to the masseswhich result in a torque being exerted on the beam, and thus to angulardeflection of the masses which can be detected with an appropriatesensor. A sensitivity of about 1 E (1 E=1 Eotvos=10⁻⁹ s⁻²) can bereached but measurement requires several hours at a single position dueto the necessity to recalculate the gravity gradient components from atleast 5 independent measurements of an angular deflection each with adifferent azimuth angle.

Practical devices, which have been built in accordance with this basicprinciple, are large in size and have low environmental noise immunity,thus requiring specially prepared conditions for measurements whichexcludes any possibility of using them on a moving carrier.

A method for absolute measurement of gravity gradient tensor componentswhich enhances the above method was invented by Forward in the middle ofthe sixties (see U.S. Pat. No. 3,722,284 (Forward et al) and U.S. Pat.No. 3,769,840 (Hansen). The method comprises mounting both a dumbbelloscillator and a displacement sensor on a platform which is in uniformhorizontal rotation with some frequency Ω about the axis of thetorsional filament. The dumbbell then moves in forced oscillation withdouble the rotational frequency, whilst many of the error sources andnoise sources are modulated at the rotation frequency or not modulated(particularly 1/f noise). The forced oscillation amplitude is at amaximum when the rotation frequency satisfies the resonance condition2Ω=ω₀, where ω₀ is the angular resonant frequency, and the oscillatorquality factor Q tends to infinity. Unlike the non-rotating method, thismethod enables one to determine rapidly the quantities Γ_(yy) -Γ_(xx)and Γ_(xy) by separating the quadrature components of the response usingsynchronous detection with a reference signal of frequency 2Ω.

The same principles can be directly used, as proposed by Metzger (seeU.S. Pat. No. 3,564,921), if one replaces the dumbbell oscillator withtwo or more single accelerometers properly oriented on such a movingplatform. There are no new features of principle in this solution tocompare with the previous one except that the outputs of the pairs ofaccelerometers require additional balancing.

Devices have been built according to this method, but they have met moreproblems than advantages, principally because of the need to maintainprecisely uniform rotation and the small displacement measurement withrespect to the rotating frame of reference. The devices have reached amaximum working accuracy of about a few tens of Eotvos for a one secondmeasuring interval, and they are extremely sensitive to environmentalvibrational noise due to their relatively low resonant frequencies. Thetechnological problems arising in this case are so difficult to overcomethat the existing developed designs of rotating gravity gradiometers areso far only at the stage of prototypes whose measurement accuracy ismuch lower than the limiting theoretical estimates.

In a paper by A. Nicolaidis and A. Taramopoulos (Il Nuovo Cimento, Vol.107B, N.11, pages 1261-1266, November 1992), the theoretical motion of astring with fixed ends under the influence of a plane monochromatictime-varying gravitational wave is discussed. According to thisdocument, a string with fixed ends may be excited to resonance providedcertain conditions, dependent on the length and orientation of thestring and the wavelength of the gravitational wave, are met. It issuggested that Fourier analysis of the motion of the string could beused to extract the direction and energy of the incident wave. Thedocument specifically avoids any discussion of the technicalimplementation of the theory, but it does suggest that strings a fewmeters or a few kilometers long should be used for the detection ofcosmological radiation or gravitational radiation from a black hole orsupernova, as the length of the string should be comparable to thewavelength of the gravitational waves. For the theoretical detector towork requires the gravitational field to oscillate in the form of agravitational wave, which would not be the case for the gravitationalfields associated with massive bodies such as the Earth.

Superconducting gravity gradiometers are known (see U.S. Pat. No.4,841,772 and Australian patent application 48185/90) utilizing a pairor more of sufficiently separated superconducting accelerometers. Evenafter greatly reducing the intrinsic and environmental thermal noisefactor, using stable persistent super-currents to balance the outputs ofthe accelerometers, and the most sensitive displacement sensors based onSQUIDs (Superconducting Quantum Interference Devices), they cannotmeasure the gravity gradient tensor components in their absolute unitsbecause they are incapable of fixing a position of the accelerometer'sproof mass which is free of all forces. Therefore, only relativedisplacements of the proof masses can be measured. Rotating designs ofsuch superconducting gravity gradiometers are not known.

Patent Abstracts of Japan vol. 009 No. 117 (P-375) and JP-A-60 050476disclose a device for measuring the acceleration due to gravity, whereina weight is suspended from a string. A current passing through thestring causes the string to vibrate in the magnetic field of a permanentmagnet. An amplified electrical signal corresponding to this vibrationis fed back to the string and the string oscillates underself-excitation at a set frequency. The acceleration due to gravity ismeasured from this frequency.

It is an object of the present invention to provide an apparatus for themeasurement of gravitational fields with improved sensitivity,portability and noise immunity over the above known systems.

It is a further object of the present invention to provide a novelapparatus for the absolute measurement of off-diagonal components of thegravity gradient tensor, in which the effect of rotation is replaced byparametric interaction between the sensitive element and active forcefeed-back connections, whereby enhanced sensitivity and vibrationalnoise immunity are attained.

It is another object of the present invention to provide a simpletechnological realisation of the above apparatus utilising theadvantages of the standard superconducting techniques which have shownan ability to reach a maximum sensitivity for mechanical displacementmeasurements and to keep intrinsic noise at a minimum level.

To achieve these objects the present invention provides apparatus forthe measurement of quasi-static gravitational fields, comprising: astring held under tension; and output means for producing an outputwhich is a function of the gravitational field, characterised over thedisclosure of Patent Abstracts of Japan vol. 009 No. 117 (P-375) andJP-A-60 050476 in that: the string is fixed at both ends; the apparatuscomprises sensing means for detecting the transverse displacement ofsaid string from an unperturbated position due to a gravitational fieldacting on said string; and the output means are responsive to thedetected displacement to produce said output which is a function of thegravitational field.

By "string" no particular limitation as to material or construction isintended. Any elongate tension element is included which is capable ofbeing transversely deflected by a gravitational field and of providing arestoring force.

An unperturbated stretched flexible string with fixed ends forms anabsolute straight line in space going through the points where the endsof the string are fixed. This line can be identified as one of the axesof the local coordinate frame, say Z, and the other two axes, X and Y,are chosen to lie in the transverse (to the string) plane. Any stringdeflection from this line is caused by absolute values of the transversecomponents of the force per unit length which is applied to each unitelement of the string.

Viewed from another aspect the invention provides a method of measuringquasi-static gravitational fields, comprising: providing a string heldunder tension; producing an output, said output being a function of saidgravitational field, characterised in that: the string has fixed ends;the method further comprises detecting the transverse displacement ofsaid string from an unperturbated position due to a gravitational fieldacting on said string; and the output which is a function of thegravitational field is produced in response to the detecteddisplacement.

The string's deflection from its unperturbated position can be easilydetected, by any suitable displacement sensing device.

Preferably the string is formed of conductive, most preferablysuperconductive material. In this case, if an electric current flowsthrough the string, a magnetic field distribution is produced in thetransverse plane and along the string's direction. If the string is madeof a superconducting material, a maximum current can be carried, and aconsequent maximum sensitivity to the deflection can be reached. A d.c.or an a.c. current may be produced in the string by incorporating thestring into a current-carrying circuit directly or by an inductivecoupling with a pumping circuit(s), provided that the string forms partof a closed conducting or superconducting loop. An a.c. current may beinduced in the string, for example by means of one or more, preferablylongitudinally symmetrically positioned, coils, which may possibly besuperconducting. The use of an a.c. current is advantageous in that itallows synchronous detection of the output signal.

When the string carries a current, the transverse magnetic field aroundthe string may interact with other conductors, or superconductors, byinductive coupling. The amplitude of the current induced in a conductoradjacent the string will be directly related to the distance of thestring from that conductor. Thus, in a preferred embodiment of theinvention one or more fixed pick-up coils are arranged along the lengthof the string to act as displacement sensing means, the current inducedin each coil being directly related to the string's displacement fromits unperturbated position.

In a preferred embodiment of the invention the sensing means comprisesat least two sensors, possibly pick-up coils, positioned symmetrically,in the longitudinal direction, with respect to the mid point of thestring.

In a particularly advantageous embodiment, displacement sensors, forexample pick-up coils, are arranged adjacent the string in twonon-parallel preferably orthogonal, planes, so as to be capable ofmeasuring the string's displacement in two transverse directionssimultaneously.

It will be understood that the displacement of a string of length l fromits unpertubated position, for example, in the y-direction of the abovelocal coordinate frame as a function of the z-position of a unit elementand time, y(z,t), can be described by the following differentialequation ##EQU1## with boundary conditions corresponding to the fixedends of the string, i.e. y(0,t)=y(1,t)=0. In this equation η denotes thestring's mass per unit length, h is the friction coefficient per unitlength, the parameters Y, A and Δl/1 are the string's Young modulus, thearea of its cross section and the string's strain respectively. Thequantities g_(y) (0,t) and Γ_(yz) (0,t) are the absolute values of they-component of the total acceleration and the corresponding gravitygradient tensor component along the string, both taken at the center ofthe local coordinate frame chosen. The function f_(L) (z,t) representsthe Langevin random force per unit length acting on the string due toits interaction with the thermostat having the absolute temperature T,with the following correlation function

    f.sub.L (z,t)f.sub.L (z',t')=4k.sub.B Thδ(z-z')δ(t-t')(2)

where k_(B) =1.4 10⁻²³ JK⁻¹ is the Boltzmann constant and δ(x-x') is thedelta-function.

In this description, the y-direction has been chosen as an arbitraryexample to simplify the explanation of the invention. However, theforegoing and following analysis is equally applicable to any directiontransverse to the string or any number of directions.

Applying Fourier analysis to the complex shape of the string caused byits interaction with the gravitational field, the function y(z,t), canbe described, in the range z=0 to z=1, by an infinite sum of sinusoidalfunctions of period 2l, with appropriate coefficients c_(y) (n,t). Thusa solution of Eq.(1), which satisfies the boundary conditions shownabove, can be represented by the following sum wherein each term in ncorresponds to one of the string's natural vibrational modes ##EQU2## Bysubstituting Eq.(3) into Eq.(1) and by multiplying its left-hand andright-hand sides by sin(πn'z/1), and then by integrating both sides overz from 0 to 1, one can obtain the differential equation for c_(y) (n,t)##EQU3## where the quantities ##EQU4## represent the string's naturalfrequencies; τ and ρ are the relaxation time and the volume mass densityof the string respectively.

When n takes an even value, i.e. for those terms of the infinite sum inEq. 3 corresponding to vibrational modes of the string having a node atz=1/2 (antisymmetric modes), the term involving g_(y) (0, t) is equal tozero. Thus, for n even, c_(y) is dependent only on Γ_(yz) (and thermalnoise).

In practice this means that the amplitude, c_(y), of the antisymmetricsinusoidal components of the deflection of the string in they-direction, y(z,t), is dependent only on the magnitude of the gravitygradient tensor component Γ_(yz).

The mid point of the string, z=1/2, is the position of a node in allantisymmetric vibrational modes of the string. If sensors are positionedsymmetrically in the longitudinal direction with respect to this point,it will be possible to identify displacements of the stringcorresponding to the string's natural antisymmetric vibrational modeswhile discounting displacements corresponding to symmetric modes, themagnitude of which is not only affected by the gravity gradient tensorcomponent Γ_(yz) but also the absolute acceleration due to gravity inthe y-direction, g_(y).

It is particularly advantageous if displacement sensors are positionedat z=1/4 and z=31/4, positions corresponding to the antinodes of thefirst antisymmetric vibrational mode of the string, n=2. At these pointsthe displacement of the string corresponding to the n=2 mode is at amaximum and thus the sensing signal will also be at a maximum, givingoptimum sensitivity.

According to a further development of the invention a conductor may beprovided adjacent the conductive string. The conductor may carry acurrent directly related to the output of the sensing means, by the useof a positive feedback loop. The current may be activated continuouslyor periodically, for example in an "off-on" manner. In this case, asmall deflection of the string due to a gravitational field will beamplified by the magnetic interaction of the string and conductor. Inother words, the conductor will "push" (or "pull") the string intofurther deflection in direct response to a small deflection caused by agravitational field acting on the string. This is clearly advantageousin that the displacement of the string is greater by virtue of themagnetic interaction with the conductor and is therefore more readilymeasurable, improving the sensitivity of the apparatus.

In a particularly advantageous embodiment of this development, two ormore conductors, possibly superconductors, are positioned longitudinallysymmetrically about the mid-point of the string so that they amplify theantisymmetric modes of the string.

In overview, a preferred embodiment of the invention provides a novelapparatus for measuring absolutely and simultaneously a pair ofoff-diagonal components of the gravity gradient tensor by means of aflexible superconducting current-carrying string with fixed ends,comprising active parametric force feed-back connections. The string isthe coherent sensitive element whose symmetric natural transverse modesare caused by the total acceleration in the transverse plane, whilst theantisymmetric modes are caused only by absolute values of the gravitygradient components along the string's direction.

In this embodiment the string forms a low-inductance part of a closedsuperconducting loop which is inductively coupled to a high-inductancedriving solenoid(s) carrying an a.c. reference current from an externalpumping source with some frequency Ω. The string is also inductivelycoupled to two superconducting magnetic flux transformers eachcomprising a signal coil and two pick-up coils wherein the pairs ofpick-up coils lie in two perpendicular planes the cross-line of whichcoincides with the unperturbated string, thus forming two independentchannels of measurements. The two arms of each superconducting fluxtransformer are balanced to convert only the string's antisymmetricnatural modes into signal current in the signal coil to be measured withSQUID's (Superconducting Quantum Interference Devices) electronics. Theoutput voltage of each channel, deeply modulated with the frequency Ω,is then proportional to the amplitudes of the antisymmetric naturalmodes of the string. This voltage is passed through a differentiatingand summing amplifier, and then used to load the feed-back circuit toproduce an in-line feed-back current distribution proximate and parallelto the string. By adjusting the feed-back current, the effectiverelaxation time and the resonant frequency of the first antisymmetricnatural mode of the string (whose amplitude depends only upon thegravity gradient along the string's direction) can be increased anddecreased respectively, while the same parameters for the symmetricnatural modes (whose amplitudes depend upon the total acceleration inthe transverse-to-string plane) are not changed. In use, the feed-backcircuit shifts the Brownian and vibrational noise level to far below thesensitivity required for industrial applications.

A preferred embodiment of the invention will now be described by way ofexample only and with reference to the following drawings in which:

FIG. 1 is a general schematic representation of a preferred embodimentof the invention; and

FIG. 2 is a diagrammatic vertical cross-section of a device according toa preferred embodiment of the invention.

A single channel prototype of a device according to the invention (seeFIG. 2) has a flexible string 1. The string is preferably formed of asuperconducting material such as Niobium (Nb). Niobium wire is the bestchoice, having optimum elastic properties, which have been proven to beusable at 4.2 K. The string forms the low-inductance part L_(o) of asuperconducting closed loop which is inductively coupled tohigh-inductance driving solenoid(s) L_(d) carrying an a.c. referencecurrent I_(d) (t) from an external pumping source with some frequency Ω.The rest of the loop is provided by the casing of the device 2,2',3,4,5.

The string has, in this embodiment, a length l=24 cm, is 1 mm indiameter and is fixed at its ends by two Nb cups 2,2' of cylindricalshape each having a hole of 1 mm diameter at its centre. The cups 2,2'close tightly the ends of a Nb cylinder comprising three parts 3,4,5connected together with two Nb cylindrical rings 6,6' carrying a finethread. The parts 3 and 5 also carry threads to engage other elements ofthe construction. The string's tension is produced by two Nb nuts 7,7'of 1 mm fine thread.

The whole construction forms a closed superconducting cylindrical cavitywith the string axially positioned. There are three spaces 10, 11, 12inside this volume electromagnetically insulated as much as possiblefrom each other by Nb partitions 9. In two of them 10, 12, drivingtoroidal solenoids L_(d2) and L_(d2), wound with 0.01 mm Nb wire andconnected in series, are placed, thus forming a large mutual inductanceM_(d) between L_(d) =L_(d1) +L_(d2) and the inductance of thecylindrical cavity L_(o) which is of the order 10⁻⁷ H for the sizeschosen. The ratio M_(d) /L_(o) is about 5×10², so if the a.c. pumpingcurrent I_(d) (t) in the driving solenoids has an amplitude of about 100mA then the induced a.c. supercurrent I_(o) carried by the string isabout 50 A peak to peak. In this case, the corresponding circularcomponent of the magnetic induction B at the string's surface is nearly200 Gauss which is approximately four times smaller than the Niobiumfirst critical field B_(c1).

The two rectangular-type pick-up coils L_(p1) and L_(p2) of thesuperconducting flux transformer, and the active force feed-back circuitare mounted together on a Titanium tube 8 placed inside the centralspace of the construction shown in FIG. 2. Titanium is chosen because ithas a thermal expansion coefficient matching that of Niobium. The activeforce feed-back circuit comprises two arms of 0.5 mm insulated copperwire stretched parallel to the string and carrying the feed-back currentI_(y) =I_(y1) +I_(y2).

This design has particular advantages; for instance, the closedsuperconducting configuration gives optimum shielding against externalvarying electromagnetic fields. Also, the cylindrically symmetricconfiguration has a small radial size which, including all integralparts of the prototype, is no more than 3.8 cm diameter. Thus, it ispossible to utilize a standard commercial 100 liter liquid helium vesselhaving an input opening of about 4 cm diameter to cool the constructiondown with a standard probe. Special helium cryostats, which have beenused for known devices, exclude the possibility of removing the devicefrom the cryostat's inner volume, for example if something goes wrong,to readjust it under field conditions. Removal of a device from acryostat requires a long period, of for example several hours, to warmthe cryostat contents to atmospheric temperature so that the contents donot explode under rapid thermal expansion. This is one of the majordisadvantages of known constructions. However, the small input openingof the standard commercial 100 liter liquid helium vessel prevents suchan explosion occurring, which means that the apparatus according to thepresent invention can be removed from the vessel and adjusted underfield conditions.

The string is deflected by a non-uniform quasi-static gravitationalfield and interacts with a variable feed-back current distributed closeto and substantially parallel to the string. The distribution is optimumwhen the feed-back current is injected at or taken from the point in thefeed-back circuit opposite the mid-point of the string (see FIG. 1). Inthe local coordinate frame chosen this point is z=1/2. Anotherrequirement for optimum operation is that the two arms of the feed-backcircuit are substantially equal and grounded at their ends. In thiscase, there is no electromagnetic coupling between the feed-back currentand the closed superconducting loop in which the string is incorporated.

The current I_(o) (t) flowing through the string and interacting withthe feed-back current distribution I_(y) (z,t) gives rise to thefollowing transverse component of the force per-unit-length f_(y) (z,t)acting on the string ##EQU5## where μ₀ =4π 10⁻⁷ Hm⁻¹ is the magneticvacuum permeability, d is the distance between the center of theunperturbated string and the center of the wire carrying the feed-backcurrent and the phase of the pumping current source 15 is chosen to bezero. The sign + or - is determined by the output buffer of thedifferentiating and summing amplifier 16 shown in FIG. 1. The transversemotion of the unit element of the string in the OYZ plane is thendescribed by the following differential equation ##EQU6## which will beseen to correspond to Eq.(1) with the addition of the term of Eq.(6).Consequently, Eq.(7) also has solutions of the form of Eq.(3). Thus,following the same algebraic manipulation as for Eq.(1), a differentialequation for c_(y) (n,t) can be obtained for this embodiment. This isEq.(8) which corresponds to Eq.(4) but with the addition of a feedbackterm. ##EQU7## The quantity ε_(n) relates to the characteristics of thetransducer system of the feedback loop; the longer the length of thearms of the feedback circuit, the larger the quantities ε_(n) are. Ifthe arms of the feed-back circuit are absolutely identical then thequantities ε_(n) are equal to zero for all odd n=1,3,5 . . . . Theirparticular values for the sizes shown in FIG. 2 are determined by##EQU8## So, for the properly adjusted feed-back circuit, only theantisymmetric natural modes of the string interact with the feed-backcurrent. However, only the antisymmetric natural modes of the string aresensitive to the absolute value of the gravity gradient tensor componentto be measured, as is seen from Eq.(4) and (7).

The superconducting pick-up coils L_(p1) and L_(p2) are placed near thestring and cause two arms of the superconducting magnetic fluxtransformer to convert, if perfectly balanced, only the antisymmetricnatural modes into the signal current I_(i) to be detected with theSQUID's electronics 13 (see FIG. 1). One uses SQUID's (SuperconductingQuantum Interference Devices) 13 as they are the most sensitive variablecurrent and magnetic flux sensors currently available. In the prototypeshown in FIG. 2, the pick-up coils are made in the form of tworectangular-type single loops of Nb wire placed symmetrically withrespect to the midpoint of the string and connected in parallel with thesignal coil L_(i). If the symmetry is perfect and the areas of the loopsare absolutely identical, then the symmetric natural modes do notproduce any signal current I_(i) or feed-back current I_(y). The sameeffect can be achieved for slightly non-identical pick-up coils with theaccuracy required, if one uses the additional inductance(s) L_(b)connected in parallel and/or series with one or both of the pick-upcoils. The inductance(s) L_(b) can be tuned to balance the two arms ofthe superconducting flux transformer. The residual "zero-model" currentin the signal coil L_(i) corresponding to the unperturbated position ofthe string can be compensated directly inside the SQUID by an additionalcoupling (not shown) to the pumping current source. If the balancingconditions are satisfied, then the output voltage of the SQUID'selectronics 13 is determined by ##EQU9## where K is the total flux tovoltage transfer function and L_(s) is the SQUID's inductance. Thequantities β_(n) depend on the physical design and position of thepick-up coils and are equal to zero if n=1,3,5 . . . . The functionΦ_(N) (t) is the equivalent-to-noise random magnetic flux inside theSQUID loop, whose spectral density S.sub.Φ (Ω) determines the intrinsicinstrumental limit of the accuracy of measurements. The feed-backcurrent I_(y) (t) is formed from the output voltage V_(y) (t) by passingit through a differentiating and summing amplifier 16 which is loaded byresistance R_(y). In this case, the feed-back current I_(y) (t) can berepresented by ##EQU10## where p, q and τ* are constant parameters whichdepend upon the design of the differentiating and summing amplifier 16.

It must be noted that a mismatch between the two arms of the feed-backcircuit always exists. The design shown in FIG. 2 uses two identicalfeed-back resistances, R_(y1) and R_(y2), one for each arm. In this casethe mismatching can be easily compensated by tuning one of theresistances, say R_(y2), to obtain the optimum case.

Equations (7) and (11) represent a closed infinite set ii ofdifferential parametric-type equations. Careful analysis has shown thatwe can ignore the terms involving the quantities c_(y) (n,t) with n>2 inthe right-hand side of Eq.(11). The reason is that just one mode can bemade "soft" i.e. the most sensitive to the gravity gradient, namelyc_(y) (2,t). If the string's natural frequencies are high enough andseparated by single octave gaps, only second-order corrections arerequired which can be easily taken into account along with analysis ofother instrumental errors. Then, as follows from Eq. (7), theself-consistent equation for the gravity gradient sensitive mode, n=2,including unavoidable fundamental noise sources can in turn be writtenin the form ##EQU11## and it is assumed that the true sign of thefeed-back current has been chosen.

If some easily carried out conditions are satisfied, which are ##EQU12##then one can show that the self-consistent output voltage is ##EQU13##where under "brownian noise" the combination of thermal and back-actionnoises is implied.

It is of interest to estimate the limiting accuracy of measurements ofthis embodiment of the invention, which can be represented by the valueof a minimum detectable gravity gradient ##EQU14## where τ_(eff)=τ/(1-ατ/4) is the effective relaxation time, m is the total mass of thestring, and E.sub.Φ (Ω) is the energy resolution of the SQUID. Using thefollowing practical parameters: l=0.24 m, m≅1.6 10⁻³ kg, τ_(eff) ≅10⁴ s,β₂ ≅4×10³ m⁻¹, L_(s) ≅5×10⁻¹¹ H, I_(o) ≅50 A, (ω₂ ² -ω² /2)^(1/2/) 2π≅2Hz, ω₂ /2π≅40 Hz, Ω/2π≧2×10² Hz, E.sub.Φ (Ω)≅2×10⁻³¹ J/Hz (d.c. biasedSQUID), one can obtain from Eq.(16) ##EQU15## It can be shown, that arange of the parameters τ, ω₂, ω, α and Ω exists where the string'sresponse described by Eq. 12 is stable. For example, for quasistaticgravity gradients and sufficiently high pumping frequency Ω one canignore the oscillating terms containing Cos(2Ωt) and Sin(2Ωt) in theright side of Eq. 12.

There are a number of detecting strategies which can be employed by thepresent invention at this stage, which are dependent on the initialmechanical parameters of the string and the application for which theapparatus is intended. It is preferable to use a string with a highmechanical stiffness and a short relaxation time in order to increaseimmunity to vibrational noise, which is the main noise source inindustrial applications, particularly in mobile gravity gradiometry. Onthe other hand, the stiffer the string, the stronger the feedback forcethat has to be applied to the string to soften the signal mode, and thelarger the back-action noise associated with the feedback current.

Additionally, the shorter the string's relaxation time, the stronger theinfluence of thermal fluctuations of the string on the measuringaccuracy since the mass per unit length of the string will normally bequite small.

To overcome both of these problems a best mode of carrying out gravitygradient measurements according to another embodiment of the presentinvention uses variable feedbacks in an "off-on" manner. In this case,the feedback force is initially not applied to the string for an`off-period` during which the string reaches thermodynamic equilibrium.The feedback force is then quickly activated for an `on-period` duringwhich the effective natural frequency ##EQU16## and the effectiverelaxation time ##EQU17## become substantially smaller and longerrespectively compared to the corresponding initial parameters of thestring. The feedback is adjusted in such a way that the effectiverelaxation time becomes much longer than the on-period. Measurements arecarried out during the on-period only, in which the string never reachesthermodynamic equilibrium. For example, the fluctuation dissipationtheorem is no longer applicable to the string during the period ofmeasurements and its response to all external noise sources is changed(see V. B. Braginsky and A. B. Manukin, Measurement of Weak Forces inPhysics Experiments, Ed. by D. H. Douglass, University Press of Chicago,1977).

One can show that in this case the least gravity gradient detectable bythis embodiment of the invention can be represented by ##EQU18## τ_(m)is the measurement time (on-period), m is the total mass of the string,E.sub.Φ (Ω) is the energy resolution of the SQUID at the frequency Ω andδ is a statistical error of the first kind. The value of δ is thelikelihood that the equivalent gravity gradient noise will exceed thevalue represented by the left side of Eq. 20 for the period ofmeasurement.

Using the following practical parameters: l=0.24 m, M=1.6×10⁻³ kg, τ=0.5s, τ_(m) =1 s, τ_(eff) ≅10⁴ s, β₂ ≅4×10³ m⁻¹, L_(S) ≅5×10⁻¹¹ H, I_(o)≅50 A, ω_(eff) /2π≅3 Hz, ω₂ /2π≅80 Hz, Ω/2π≧10⁴ Hz, E.sub.Φ (Ω)≅5×10⁻³²J/Hz (500 d.c. biased SQUID), one can obtain from Eq.(20)

    Γ.sub.min =0.02 Eotvos

In both the above embodiments, the desired signal is obtained from theoutput voltage by synchronous detection with a reference signal takenfrom the pumping source 15, and the invention allows calibration of thedesired signal in gravity gradient absolute units without rotation ashas been proposed for known rotating gravity gradiometers. As forrotating designs, the invention allows the movement of the noisespectrum to a frequency range at which 1/f contribution is sufficientlysmall. Natural vibrations of the string, which occur during the time ofmeasurement (on-period), do not cause a problem since they can befiltered out from the desired signal provided that the on-period ischosen to be much longer than the period (2π/ω_(eff)) of suchvibrations.

Vibrational noise immunity is improved by the factor (ω_(eff) /ω₁)²which can be made as small as 10⁻².

One must consider inductive cross-coupling between the feedback currentsand each pair of the pick-up coils and cross-coupling between thepick-up coils themselves, both of which act like negative feedbackloops. On the one hand this leads to unnecessary renormalisation of theamplitudes of the output signals until the gain of the SQUID'selectronics exceeds some critical value. On the other hand in the caseof double-channel measurements, the output signal of each channelcontains a linear combination of each gravity gradient component to bemeasured. It can be shown that each of such components can,nevertheless, be measured separately and simultaneously, if a properdata acquisition system is used. The effect can be easily eliminated byorganising additional positive feedback to counteract this negativefeedback, for example by connecting, via a weak inductive coupling, eachfeedback current with each SQUID.

In practice, the apparatus according to the invention can be used todetermine in absolute units the off-diagonal components of the gravitygradient. By conducting a gravity survey over an area, small differencesin absolute gravity gradient can be detected. Such small changes mayindicate variations in local geological features, for example, thepresence of minerals, gas or oil.

Repeated readings over time at a single locality could indicate changinggeological status of an area, such as rising magma. Clearly theinvention enhances prospecting and other data gathering pursuits whereaccurate gravitational field measurement is required. Use of absolutevalues enhances the information that can be determined from the datameasured. A gradiometer according to the invention can be used whilemoving, which allows the gradiometer to be used on vehicles whetherland, sea or air vehicles. For example, the device can be suspended froma helicopter and used while the helicopter traverses a selected area.

I claim:
 1. Apparatus for the measurement of quasi-static gravitationalfields, comprising:a string composed of conductive material, fixed atboth ends, held under tension and arranged to carry a current I_(o) ;sensing means for detecting the transverse displacement of said stringfrom an unperturbated position due to a gravitational field acting onsaid string; and means responsive to the detected displacement forproducing an output which is a function of the gravitational field. 2.Apparatus as claimed in claim 1, wherein said sensing means comprises atleast two sensors symmetrically longitudinally positioned about themid-point of said string.
 3. Apparatus as claimed in claim 1 furthercomprising conductive means adjacent said string carrying a currentI_(y), wherein the magnitude of the current I_(y) is a function of theoutput of the sensing means; and the magnetic fields associated with thecurrent I_(y) through the conductive means and the current I_(o) throughthe string interact to produce a feedback force on said string so as toincrease the transverse displacement of the string from itsunperturbated position in response to the gravitational field acting onsaid string.
 4. Apparatus as claimed in claim 3, wherein said conductivemeans comprises at least two conductors longitudinally symmetricallypositioned about the mid-point of said string, each conductor carrying asubstantially equal proportion of said current I_(y).
 5. Apparatus asclaimed in claim 3 wherein the current I_(y) through said conductivemeans is activated periodically.
 6. Apparatus as claimed in claim 1,wherein said sensing means comprises at least one pick-up coil in whicha current I_(p) is induced by said current I_(o) through said string,said current I_(p) being a function of said string's displacement. 7.Apparatus as claimed in claim 1, wherein said current I_(o) through saidstring is an alternating current.
 8. Apparatus as claimed in claim 1wherein said current I_(o) is induced in said string by inductive means.9. Apparatus as claimed in claim 8, wherein said inductive meanscomprises two solenoids longitudinally symmetrically positioned aboutthe mid-point of said string.
 10. Apparatus as claimed in claim 1,wherein said string is composed of superconducting material. 11.Apparatus as claimed in claim 1, wherein said sensing means comprisesmeans for detecting the transverse displacement of said string in twonon-parallel planes.
 12. A method of measuring quasi-staticgravitational fields, comprising:providing a string composed ofconductive material with fixed ends and held under tension; passing acurrent through said string; detecting the transverse displacement ofsaid string from an unperturbated position due to a gravitational fieldacting on said string; and producing an output in response to thedetected displacement, said output being a function of saidgravitational field.
 13. A method as claimed in claim 12 wherein saidoutput is produced by measuring the spatial position of at least onepoint on the string relative to the unperturbated position of saidpoint.
 14. A method as claimed in claim 13 wherein the spatial positionsof an even plurality of points on the string are measured relative totheir unperturbated positions and said points are chosen to liepair-wise longitudinally symmetrically about the mid-point of thestring.
 15. A method as claimed in claim 14 wherein said pointscorrespond to the positions of antinodes of the antisymmetric naturalmodes of said string.
 16. A method as claimed in claim 12 wherein saiddisplacement of said string is increased by applying a feedback force tothe string, said force being a function of the gravitational fieldacting on the string.
 17. A method as claimed in claim 16 wherein saidfeedback force is a direct function of said output.
 18. A method asclaimed in claim 17 wherein said feedback force is applied to saidstring so as to accentuate the components of the spatial configurationof the string corresponding to natural antisymmetric modes of the stringin preference to the components of the spatial configurationcorresponding to natural symmetric modes.
 19. A method as claimed inclaim 12 wherein said displacement is measured in two non-parallelplanes.
 20. A method of measuring absolute off-diagonal components ofthe gravity gradient tensor, the method comprising the stepsof:providing a flexible string with fixed ends; applying a forceper-unit-length to said string; detecting the deflection of said stringfrom its unperturbated position due to absolute values oftransverse-to-string components of said force per-unit-length applied toeach unit element of said string; and analyzing the detected deflection,wherein said deflection is a combination of said string's natural modes,and the even said modes are caused only by absolute values of thecomponents of the gravity gradient in the string's direction, whilst theodd said modes are caused by total acceleration in thetransverse-to-string plane.
 21. Apparatus for measuring off-diagonalcomponents of the gravity gradient tensor of quasi-static gravitationalfields comprising:a string held under tension, said string being fixedat both ends; means for applying a force per-unit-length to said string;sensing means for detecting the transverse displacement of said stringfrom an unperturbated position due to a gravitational field acting onsaid string; and output means responsive to the detected displacementfor producing an output which is a function of the gravitational field,wherein deflection of said string is caused by absolute values oftransverse-to-string components of said force acting on said string, ina manner such that deflection of said string is a combination of saidstring's natural modes, and the even said modes are caused only byabsolute values of the components of the gravitational gradient in thestring direction, while the odd said modes are caused by totalacceleration in the transverse-to-string plane.